NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.7%

Time: 13.6s

Alternatives: 8

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot 0.5}{a + b}}{a \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ (* PI 0.5) (+ a b)) (* a b)))

C

double code(double a, double b) {
	return ((((double) M_PI) * 0.5) / (a + b)) / (a * b);
}

Java

public static double code(double a, double b) {
	return ((Math.PI * 0.5) / (a + b)) / (a * b);
}

Python

def code(a, b):
	return ((math.pi * 0.5) / (a + b)) / (a * b)

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * 0.5) / Float64(a + b)) / Float64(a * b))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * 0.5) / (a + b)) / (a * b);
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+104}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+104)
   (* PI (/ 0.5 (* a (* b (+ a b)))))
   (/ (/ (* PI 0.5) b) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+104) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6e+104) {
		tmp = Math.PI * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = ((Math.PI * 0.5) / b) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6e+104:
		tmp = math.pi * (0.5 / (a * (b * (a + b))))
	else:
		tmp = ((math.pi * 0.5) / b) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6e+104)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6e+104)
		tmp = pi * (0.5 / (a * (b * (a + b))));
	else
		tmp = ((pi * 0.5) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6e+104], N[(Pi * N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;b \leq 3.7 \cdot 10^{-68}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= b 3.7e-68) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 3.7e-68) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 3.7e-68) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if b <= 3.7e-68:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (b <= 3.7e-68)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (b <= 3.7e-68)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.7e-68], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e-68) (* (/ 0.5 (* a b)) (/ PI a)) (* (/ (/ 0.5 a) b) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e-68) {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / a);
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.4e-68) {
		tmp = (0.5 / (a * b)) * (Math.PI / a);
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.4e-68:
		tmp = (0.5 / (a * b)) * (math.pi / a)
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e-68)
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.4e-68)
		tmp = (0.5 / (a * b)) * (pi / a);
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e-68], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.3e-68) (/ PI (* a (* (* a b) 2.0))) (* (/ (/ 0.5 a) b) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.3e-68) {
		tmp = ((double) M_PI) / (a * ((a * b) * 2.0));
	} else {
		tmp = ((0.5 / a) / b) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.3e-68) {
		tmp = Math.PI / (a * ((a * b) * 2.0));
	} else {
		tmp = ((0.5 / a) / b) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.3e-68:
		tmp = math.pi / (a * ((a * b) * 2.0))
	else:
		tmp = ((0.5 / a) / b) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.3e-68)
		tmp = Float64(pi / Float64(a * Float64(Float64(a * b) * 2.0)));
	else
		tmp = Float64(Float64(Float64(0.5 / a) / b) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.3e-68)
		tmp = pi / (a * ((a * b) * 2.0));
	else
		tmp = ((0.5 / a) / b) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.3e-68], N[(Pi / N[(a * N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.3e-68) (/ PI (* a (* (* a b) 2.0))) (/ (/ (* PI 0.5) b) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.3e-68) {
		tmp = ((double) M_PI) / (a * ((a * b) * 2.0));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.3e-68) {
		tmp = Math.PI / (a * ((a * b) * 2.0));
	} else {
		tmp = ((Math.PI * 0.5) / b) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.3e-68:
		tmp = math.pi / (a * ((a * b) * 2.0))
	else:
		tmp = ((math.pi * 0.5) / b) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.3e-68)
		tmp = Float64(pi / Float64(a * Float64(Float64(a * b) * 2.0)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.3e-68)
		tmp = pi / (a * ((a * b) * 2.0));
	else
		tmp = ((pi * 0.5) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.3e-68], N[(Pi / N[(a * N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{a + b} \cdot \frac{0.5}{a \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ PI (+ a b)) (/ 0.5 (* a b))))

C

double code(double a, double b) {
	return (((double) M_PI) / (a + b)) * (0.5 / (a * b));
}

Java

public static double code(double a, double b) {
	return (Math.PI / (a + b)) * (0.5 / (a * b));
}

Python

def code(a, b):
	return (math.pi / (a + b)) * (0.5 / (a * b))

Julia

function code(a, b)
	return Float64(Float64(pi / Float64(a + b)) * Float64(0.5 / Float64(a * b)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / (a + b)) * (0.5 / (a * b));
end

Wolfram

code[a_, b_] := N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 64.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{a \cdot b} \cdot \frac{\pi}{a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ 0.5 (* a b)) (/ PI a)))

C

double code(double a, double b) {
	return (0.5 / (a * b)) * (((double) M_PI) / a);
}

Java

public static double code(double a, double b) {
	return (0.5 / (a * b)) * (Math.PI / a);
}

Python

def code(a, b):
	return (0.5 / (a * b)) * (math.pi / a)

Julia

function code(a, b)
	return Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a))
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 / (a * b)) * (pi / a);
end

Wolfram

code[a_, b_] := N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2024006 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))